On the Equivalence of Causal Models: A Category-Theoretic Approach
This provides a formal framework for comparing causal models, which is incremental as it builds on existing category-theoretic approaches in causality.
The paper tackles the problem of determining equivalence between causal models with different directed acyclic graphs by developing a category-theoretic criterion, showing that when one model is a Φ-abstraction of another, intervention calculi can be consistently translated.
We develop a category-theoretic criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. Following Jacobs et al. (2019), we define a causal model as a probabilistic interpretation of a causal string diagram, i.e., a functor from the ``syntactic'' category $\textsf{Syn}_G$ of graph $G$ to the category $\textsf{Stoch}$ of finite sets and stochastic matrices. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors, which we call a $Φ$-abstraction and $Φ$-equivalence, respectively. It is shown that when one model is a $Φ$-abstraction of another, the intervention calculus of the former can be consistently translated into that of the latter. We also identify the condition under which a model accommodates a $Φ$-abstraction, when transformations are deterministic.